Fuzzy Transforms of Higher Order Approximate Derivatives: A Theorem

Authors

Irina Perfilieva
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

12-2009

Comments

Technical Report: UTEP-CS-09-39

Abstract

In many practical applications, it is useful to represent a function f(x) by its fuzzy transform, i.e., by the "average" values over different elements of a fuzzy partition A₁(x), ..., _n(x) (for which A_i(x) >= 0 and A₁(x) + ... + A_n(x) = 1). It is known that when we increase the number n of the partition elements A_i(x), the resulting approximation get closer and closer to the original function: for each value x₀, the values F_i corresponding to the function A_i(x) for which A_i(x₀) > 0 tend to f(x₀).

In some applications, if we approximate the function f(x) on each element A_i(x) not by a constant but by a polynomial (i.e., use a fuzzy transform of a higher order), we get an even better approximation to f(x).

In this paper, we show that such fuzzy transforms of higher order (and even sometimes the original fuzzy transforms) not only approximate the function f(x) itself, they also approximate its derivative(s). For example, we have F_i'(x₀) --> f'(x₀)$.